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For a composition $\lambda$ of $n$, a Young subgroup permutes independently the blocks of a set partition of $\{1,\dots ,n\}$ with sizes ${\lambda }_i$, and is isomorphic to $S_{{\lambda }_1}\times \cdots \times S_{{\lambda }_r}$. Reordering the block sizes gives a conjugate subgroup. Inducing its trivial representation gives a Young permutation module.

A real-space renormalization group replaces groups of microscopic degrees of freedom by retained coarse variables, summing over the eliminated variables to define a coarse Hamiltonian. The effective Boltzmann weights must preserve the partition function up to a tracked normalization. Spin decimation retains selected spins rather than forming an average block variable. Coarse-graining generally generates additional interactions, so closure of a chosen finite coupling family requires justification.

A renormalizable quantum field theory permits ultraviolet divergences to be absorbed into a fixed finite set of parameters and field normalizations. In ordinary four-dimensional perturbative power counting in quantum field theory, its interaction coefficients have nonnegative mass dimension.

For nonnegative finite independent and identically distributed random variables as inter-arrivals with positive probability of exceeding $\epsilon >0$, write $p=\mathbb{P}(X_1\ge \epsilon )>0$. The count tail is bounded by $\mathbb{P}(\mathrm{Bin}(n,p)\le \lfloor t/\epsilon \rfloor )$, which has geometric decay times a polynomial in $n$. Hence $\mathbb{E}{e}^{\theta N(t)}<\infty$ for every fixed finite $t$ and sufficiently small $\theta >0$.

The Källén function controls two-body relativistic kinematics. In the rest frame of total invariant mass $\sqrt{s}$, the magnitude of each final momentum is $\sqrt{\kappa (s,m_1^2,m_2^2)}/(2\sqrt{s})$. It also determines the invariant flux factor in a two-particle collision.

Adaptive rejection sampling builds a piecewise linear upper envelope for a differentiable log-concave probability density, samples from its piecewise exponential form, and accepts with the target-to-envelope ratio. Adding evaluated points improves the envelope without changing the target distribution. Initial tangent slopes must yield integrable tails. The accepted draws are exact samples, unlike a finite run of Markov chain Monte Carlo.

Ignoring velocities and evolving potentials, a thin reionization screen transmits the primary Cosmic microwave background anisotropy with amplitude $e^{-{\tau }_{\mathrm{re}}}$ and adds a direction-averaged source. A plane wave propagating a distance $\Delta \chi$ from recombination to the screen has monopole factor $j_0(k\Delta \chi )=\sin (k\Delta \chi )/(k\Delta \chi )$. This suppresses the rescattered source at $k\Delta \chi \gg 1$, while coherent large-scale modes are unchanged to leading order.

A regularization parameter controls the tradeoff between approximation bias and data-noise amplification in an inverse problem. For early stopping of Landweber iteration with fixed step $\tau$, the convention $\alpha =1/(\tau n)$ associates a smaller parameter with more iterations.

A positive measurable function $\ell$ is slowly varying at infinity iffor every fixed $u>0$. It has regular variation of index zero. “Slowly” concerns relative change under multiplication of the argument; it does not mean that the function must be bounded, have a small derivative, or converge to a finite value.

Every positive constant is a slowly varying function. So are $\log t$, $1/\log t$ and $(\log t{)}^a$ for any fixed real number $a$, on $t>1$. Indeed,The first grows without bound, the second tends to zero, and the constant does neither. Another unbounded example is $\ell (t)=e^{\sqrt{\log t}}$:This function can grow much faster than $\log t$ while retaining regular variation of index zero.

Even convergence is unnecessary. The bounded functionoscillates between $e^{-1}$ and $e$. It is nevertheless a slowly varying function: the change in the argument of the sine under $t\mapsto ut$ isand the Lipschitz continuity of the sine makes the logarithm of the ratio tend to zero. The sequences $t_k=\exp (\exp (\pi /2+2\pi k))$ and $s_k=\exp (\exp (3\pi /2+2\pi k))$ give the two values $e$ and $e^{-1}$ for every nonnegative integer $k$, so there is no limit at infinity.

Contrast this with $h(t)=2+\sin (\log t)$, which is positive and bounded but is not a slowly varying function. Fix $u=e^{\pi /2}$. Along $t_k=e^{2\pi k}$ its rescaling ratio is $h(u{t}_k)/h(t_k)=3/2$; along $s_k=e^{\pi +2\pi k}$ the ratio is $1/2$. The persistent oscillations prevent even a limit for this one fixed rescaling. It is not regularly varying with any finite index either.

The useful distinction is whether a fixed proportional change of input eventually produces a negligible proportional change of output. A slowly varying function can therefore preserve the scaling exponent of a power law while changing its magnitude substantially. The decomposition in regular variation explains why a logarithmic factor can matter in numerical estimates without changing the limiting exponent.

For a reflexive pair $f,g:A\rightrightarrows B$ with common section $r$, form the pushout in a category of $f$ and $g$, with maps $j_1,j_2:B\to P$. The equation $j_{1}f=j_{2}g$, composed with $r$, gives $j_1=j_2=q$. This common map is a coequalizer of $f,g$: every map equalizing them defines a commuting pair into the pushout and hence factors uniquely through $q$.

A reflexive coequalizer is a coequalizer of a reflexive pair, meaning a parallel pair $r,s:X\rightrightarrows Y$ with a common section $t:Y\to X$ satisfying $rt=st=1_Y$. A functor preserves reflexive coequalizers when it takes every such coequalizer to a coequalizer of the image pair.

A nonempty norm-closed convex subset $C$ of a reflexive Banach space attains its distance to zero. For $d={\inf }_C\Vert x\Vert$, the sets $C\cap (d+1/n)B_X$ are nonempty weakly closed subsets of one weakly compact ball, and form a decreasing family. Compactness gives a point in their intersection, of norm $d$. Mazur theorem supplies weak closedness of $C$ and the balls; no sequential-compactness theorem is required.

The equal-time canonical commutation relation is $[\varphi (t,\mathbf{x}),\pi (t,\mathbf{y})]=i{\delta }^3(\mathbf{x}-\mathbf{y})$ in units $\hslash =1$, with the two field-field and momentum-momentum commutators zero. The quadratic Hamiltonian operator gives $\dot{\varphi }=\pi$ and $\dot{\pi }={\nabla }^2\varphi -m^2\varphi$, hence the Klein-Gordon equation as an operator identity.

For two free real scalar fields, the internal rotation $\delta {\varphi }_1=\theta {\varphi }_2$, $\delta {\varphi }_2=-\theta {\varphi }_1$ is a symmetry precisely when their squared masses agree. Its Noether current is $j^{\mu }={\varphi }_2{\partial }^{\mu }{\varphi }_1-{\varphi }_1{\partial }^{\mu }{\varphi }_2$. Without mass degeneracy, ${\partial }_{\mu }{j}^{\mu }=(m_2^2-m_1^2){\varphi }_1{\varphi }_2$.

A nonminimal scalar coupling includes an interaction between the field and spacetime curvature, such as $-\xi R{\Phi }^2$ in the Lagrangian density. The Euler-Lagrange field equation for $\mathcal{L}=-\frac{1}{2}(\nabla \Phi {)}^2-\xi R{\Phi }^2$ is $\square \Phi -2\xi R\Phi =0$. The coefficient is convention dependent: an action written with $-\frac{1}{2}\xi R{\Phi }^2$ has $\xi$ instead of $2\xi$.

For $m\ge 0$, the cohomology ring isandThe generator $a$ reduces to the square of the degree-one generator in the mod-two cohomology ring of real projective space; $b$ is the integral orientation class in odd top dimension.

The $n+1$ manifold charts $U_i=\{[X]:X_i\ne 0\}$ have coordinates $X_k/X_i$, with $k\ne i$. Their inverse inserts $1$ as the $i$th homogeneous coordinate. On an overlap, changing the denominator from $X_i$ to $X_j$ divides each remaining coordinate by the nonzero coordinate $X_j/X_i$. These rational smooth transition maps give Real projective space its standard smooth manifold structure.

A planar advection-driven melting front in a porous matrix with permeabilities $k$ ahead and $k+\Delta k$ behind has the ideal sharp-front growth rateThis follows by matching harmonic pressure perturbations and normal Darcy flux, then converting flux to front speed by the enthalpy balance. It predicts no finite fastest-growing wavelength when $\Delta k>0$.

Across a jump of $U^{\prime }$ in a constant-density parallel flow with continuous $U$, the normal mode vertical velocity and pressure are continuous. The linearized momentum equation gives $\hat{p}=\rho [(U-c){\hat{w}}^{\prime }-U^{\prime }\hat{w}]/(ik)$, yielding the displayed conditions. Equivalently, away from $U=c$, $(U-c)[{\hat{w}}^{\prime }]=[U^{\prime }]\hat{w}$. Derivative continuity is generally incorrect when the background vorticity jumps. This is the constant-density, continuous-velocity specialization of jump conditions for stratified inviscid shear flow.

A stochastic network combines resource constraints, routing, and random arrivals or service requirements. Loss networks reject requests that cannot acquire all required resources; flow-level network models let ongoing transfers share capacities.